[Une application de type Hénon cubique dans le déploiement d'une orbite homocline dégénérée avec résonance]
Nous étudions le déploiement d'un champ de vecteurs sur qui possède une orbite homocline dégénérée associée à une singularité hyperbolique. La partie linéaire du champ en cette singulartité posssède une résonance mais, pour le système initial, le terme résonant associé à cette résonance disparaît. Nous montrons qu'après changement d'échelle, l'application de retour de Poincaré sur une section transverse est proche d' une application de Hénon cubique. Un attracteur étrange est présent et persiste au sens de la mesure de Lebesgue. Nous montrons également la présence d'un attracteur avec une entropie topologique proche de .
In this Note, we study the unfolding of a vector field that possesses a degenerate homoclinic (of inclination-flip type) to a hyperbolic equilibrium point where its linear part possesses a resonance. For the unperturbed system, the resonant term associated with the resonance vanishes. After suitable rescaling, the Poincaré return map is a cubic Hénon-like map. We deduce the existence of a strange attractor which persists in the Lebesgue measure sense. We also show the presence of an attractor with topological entropy close to .
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@article{CRMATH_2005__340_11_843_0,
author = {Marco Martens and Vincent Naudot and Jiazhong Yang},
title = {A cubic {H\'enon-like} map in the unfolding of degenerate homoclinic orbit with resonance},
journal = {Comptes Rendus. Math\'ematique},
pages = {843--846},
publisher = {Elsevier},
volume = {340},
number = {11},
year = {2005},
doi = {10.1016/j.crma.2005.04.001},
language = {en},
}
TY - JOUR AU - Marco Martens AU - Vincent Naudot AU - Jiazhong Yang TI - A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance JO - Comptes Rendus. Mathématique PY - 2005 SP - 843 EP - 846 VL - 340 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2005.04.001 LA - en ID - CRMATH_2005__340_11_843_0 ER -
%0 Journal Article %A Marco Martens %A Vincent Naudot %A Jiazhong Yang %T A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance %J Comptes Rendus. Mathématique %D 2005 %P 843-846 %V 340 %N 11 %I Elsevier %R 10.1016/j.crma.2005.04.001 %G en %F CRMATH_2005__340_11_843_0
Marco Martens; Vincent Naudot; Jiazhong Yang. A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance. Comptes Rendus. Mathématique, Volume 340 (2005) no. 11, pp. 843-846. doi : 10.1016/j.crma.2005.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.001/
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